The fallen pine

Two congruent isosceles triangles and three lines. Prove triangle AEF is also isosceles.

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Solution by Herbert Kainberger.

Triangles ABD and CBE are similar since they are both isosceles and they have the same top angle at vertex B. Therefore also ∠EAF equals ∠AEF and hence △AEF is isosceles.

An interesting finding by Ignacio Larrosa Cañestro is that the locus of point F seems to be a hyperbola.


The fallen pine
the tree is sad
he is just be cut
by human hands
like a giant
his crown to earth
he is broken and dirty
no more rain
no more life
no more thunder
just a yearning
just a burning
a windswept pine tree
when the pine tree fell
just a big sadness


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2 replies on “The fallen pine”

In the picture below I show the formula for F if AB is fixed and h is variable.
The graph is no hyperbola.

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